84 XII TOPOLOGY AND TOPOLOGICAL ALGEBRA of a single point is nowhere dense unless it is open, i.e., unless a is isolated in E (3.10.10). In a topological group G, every closed subgroup H which is not open is nowhere dense (12.8.7). In a topological space E, a subset B is said to be meager if it is the union of at most denumerably many nowhere-dense sets. Since the closure of a nowhere- dense set is nowhere dense by definition, Baire's theorem can be restated in the form that, in a space E satisfying the conditions of (12.16.1), the complement of a meager set is dense in E. For example, in a Frechet space E, a denumerable union of closed linear varieties, each distinct from E, i.e., sets of the form an + Vrt, where Vn ^ E is a closed vector subspace of E, has a dense complement in E. For it follows from (12.13.1) that a vector subspace V of a topological vector space E cannot be open in E unless V = E. In particular, a denumerable set in a Frechet space is meager, but it should be noted that such a set may also be dense (if the space is separable). (12.16.2) Let Ebea topological space in which every point has a neighborhood homeomorphic to a complete metric space, and let u be a lower semicontinuous function on E. Ifu(x) < -|- oo for each x e E, then given any nonempty open set U ITJ E there exists a nonempty open set V c U such that sup u(x) < + oo. xeV It is enough to prove the result when U = E. For each integer n > 0, let Fn be the set of all x e E such that u(x) £ n. By hypothesis, FM is a closed set (12.7.2), and E is the union of the Fn; hence at least one of the Fw is not no- where dense (12.16.1), and therefore contains a nonempty open set. Q.E.D. Notice that under the hypotheses of (12.16.2) it can happen that sup u(x) =4-00. xeE Consider for example the real-valued function on R which is equal to 0 at x - 0, and equal to l/x2 when x ^ 0. In particular: (12.16.3) In a Frechet space E, every lower semicontinuous seminorm is con- tinuous. By definition, a seminorm/? on E is finite at all points of E. Ifp is lower semicontinuous, it follows from (12.16.2) that there exists a point x0 e E, ace or a locally compact metrizable space (by virtue